First slide

Applications of SHM

Question

The mass and diameter of a planet are twice that of the earth. What will be the time period of oscillation of a pendulum on this planet, if it is a seconds pendulum on earth

Moderate

A

$\sqrt{2}$ seconds
B

2 seconds
C

$\frac{1}{\sqrt{2}}$ seconds
D

$2 \sqrt{2}$ seconds

Solution

$a c c e l e r a t i o n d u e t o g r a v i t y o n E a r t h i s g_{e} = \frac{GM}{R^{2}} a c c e l e r a t i o n d u e t o g r a v i t y o n p l a n e t i s g_{p} = \frac{{GM}_{p}}{{R_{p}}^{2}} G i v e n r a d i u s o f p l a n e t R_{p} {=2R}_{e a r t h} {=2R}_{e} g i v e n m a s s o f p l a n e t M_{p} {=2M}_{e a r t h} {=2M}_{e}$
$\frac{g_{p}}{g_{e}} = \frac{{GM}_{p}}{{R_{p}}^{2}} × \frac{{R_{e}}^{2}}{{GM}_{e}} s u b s t i t u t e v a l u e s o f R_{p} = 2 R_{E}, M_{P} = 2 M_{e} \frac{g_{p}}{g_{e}} = \frac{{2R}_{e}^{2}}{{4R}_{e}^{2}} = \frac{1}{2} g_{p} = \frac{g_{e}}{2} ........(1) \frac{T_{e}}{T_{p}} = \sqrt{\frac{g_{p}}{g_{e}}} = \sqrt{\frac{g_{e} /2}{g}} = \frac{1}{\sqrt{2}} s u b s t i t u t e t i m e p e r i o d o n e a r t h = 2 t h a t o f s e c o n d s p e n d u l u m \frac{2}{T_{p}} = \frac{1}{\sqrt{2}} \Rightarrow T_{p} = 2 \sqrt{2} s e c o n d s .$

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